Atmosphere Impact Losses
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Schlichting, Hilke E. and Sujoy, Mukhopadhyay."Atmosphere Impact
Losses." Space Science Reviews 214 (February 2018): 34 © 2018 The
Author(s)
As Published
http://dx.doi.org/10.1007/s11214-018-0471-z
Publisher
Springer-Verlag
Version
Final published version
Citable link
http://hdl.handle.net/1721.1/115344
Terms of Use
Creative Commons Attribution
https://doi.org/10.1007/s11214-018-0471-z
Atmosphere Impact Losses
Hilke E. Schlichting1,2 · Sujoy Mukhopadhyay3
Received: 15 November 2016 / Accepted: 11 January 2018 / Published online: 23 January 2018 © The Author(s) 2018. This article is published with open access at Springerlink.com
Abstract Determining the origin of volatiles on terrestrial planets and quantifying
atmo-spheric loss during planet formation is crucial for understanding the history and evolution of planetary atmospheres. Using geochemical observations of noble gases and major volatiles we determine what the present day inventory of volatiles tells us about the sources, the ac-cretion process and the early differentiation of the Earth. We further quantify the key volatile loss mechanisms and the atmospheric loss history during Earth’s formation. Volatiles were accreted throughout the Earth’s formation, but Earth’s early accretion history was volatile poor. Although nebular Ne and possible H in the deep mantle might be a fingerprint of this early accretion, most of the mantle does not remember this signature implying that volatile loss occurred during accretion. Present day geochemistry of volatiles shows no evidence of hydrodynamic escape as the isotopic compositions of most volatiles are chondritic. This suggests that atmospheric loss generated by impacts played a major role during Earth’s for-mation. While many of the volatiles have chondritic isotopic ratios, their relative abundances are certainly not chondritic again suggesting volatile loss tied to impacts. Geochemical evi-dence of atmospheric loss comes from the3He/22Ne, halogen ratios (e.g., F/Cl) and low H/N
ratios. In addition, the geochemical ratios indicate that most of the water could have been delivered prior to the Moon forming impact and that the Moon forming impact did not drive off the ocean. Given the importance of impacts in determining the volatile budget of the Earth we examine the contributions to atmospheric loss from both small and large impacts. The Delivery of Water to Protoplanets, Planets and Satellites
Edited by Michel Blanc, Allessandro Morbidelli, Yann Alibert, Lindy Elkins-Tanton, Paul Estrada, Keiko Hamano, Helmut Lammer, Sean Raymond and Maria Schönbächler
B
H.E. SchlichtingS. Mukhopadhyay
1 University of California, Los Angeles, 595 Charles E. Young Drive East, Los Angeles, CA 90095-1567, USA
2 Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
We find that atmospheric mass loss due to impacts can be characterized into three different regimes: 1) Giant Impacts, that create a strong shock transversing the whole planet and that can lead to atmospheric loss globally. 2) Large enough impactors (mcap
√
2ρ0(π hR)3/2, rcap∼ 25 km for the current Earth), that are able to eject all the atmosphere above the
tan-gent plane of the impact site, where h, R and ρ0are the atmospheric scale height, radius of
the target, and its atmospheric density at the ground. 3) Small impactors (mmin>4πρ0h3,
rmin∼ 1 km for the current Earth), that are only able to eject a fraction of the atmospheric
mass above the tangent plane. We demonstrate that per unit impactor mass, small impactors with rmin< r < rcapare the most efficient impactors in eroding the atmosphere. In fact for
the current atmospheric mass of the Earth, they are more than five orders of magnitude more efficient (per unit impactor mass) than giant impacts, implying that atmospheric mass loss must have been common. The enormous atmospheric mass loss efficiency of small im-pactors is due to the fact that most of their impact energy and momentum is directly available for local mass loss, where as in the giant impact regime a lot of energy and momentum is ’wasted’ by having to create a strong shock that can transverse the entirety of the planet such that global atmospheric loss can be achieved. In the absence of any volatile delivery and outgassing, we show that the population of late impactors inferred from the lunar cra-tering record containing 0.1% M⊕is able to erode the entire current Earth’s atmosphere implying that an interplay of erosion, outgassing and volatile delivery is likely responsible for determining the atmospheric mass and composition of the early Earth. Combining geo-chemical observations with impact models suggest an interesting synergy between small and big impacts, where giant impacts create large magma oceans and small and larger impacts drive the atmospheric loss.
Keywords Volatiles· Accretion · Planet formation · Water · Impacts
1 Introduction
Terrestrial planet formation can generally be characterized by three distinct stages. The first is dominated by the accretion of planetesimals, which results in the formation of planetary embryos comparable to their isolation masses. The isolation mass is simply the mass that a planetary embryo can grow to by locally accreting all the material available in its feeding zone and it is given by
Miso=
(10π Σa2)3/2
(3M)1/2 , (1)
where Σ is the disk surface density in solids, a the semi-major axis and Mthe mass of the sun (e.g. Ida and Makino1993; Weidenschilling et al.1997; Schlichting2014). Evaluating the isolation mass, assuming that Σ is given by the minimum-mass solar nebular (MMSN),
ΣMMSN 10 × (a/1 AU)−3/2g cm−2 (Hayashi1981), yields at 1 AU Miso∼ MMars. This
implies that Mars may indeed have formed and survived as an isolation mass, but that Earth and Venus must have undergone an additional stage of accretion. The second stage consists of a series of giant impacts of a few dozen embryos that merge to form the Earth and Venus (e.g. Agnor et al.1999; Chambers2001), with the Moon forming impact being the last gi-ant impact of the Earth (e.g. Cameron and Benz1991; Canup and Asphaug2001; ´Cuk and Stewart2012). Giant impacts begin when the planetesimals are no longer able to efficiently damp the eccentricities that are mutually excited by the growing planetary embryos. Order of magnitude estimates that balance the stirring rates of the protoplanets with the damping
rates due to dynamical friction by the planetesimal population and numerical simulations find that giant impacts set in when the total mass in protoplanets is comparable to the mass in planetesimals (Goldreich et al.2004; Kenyon and Bromley2006; Schlichting et al.2012). This implies that about 50% of the total mass still resides in planetesimals when giant im-pacts begin and planetesimal accretion continues throughout the giant impact phase and afterwards. Furthermore, there exists geochemical evidence that a total of about 0.01 M⊕ of chondritic material was delivered as ‘late veneer’ by planetesimals to the terrestrial plan-ets after the end of giant impacts (Warren et al.1999; Walker et al.2004; Walker2009), which is the third stage of accretion. Such late veneer, if accreted in small planetesimals, has the additional advantage that it can damp, due to the absence of strong mutual excitation among the terrestrial planets by the end of the impact stage, the post giant impact eccen-tricities to the almost circular values that we observe today (Schlichting et al.2012). The terrestrial planets therefore had a rich and diverse formation history with volatiles being de-livered and eroded during the planets’ accretion of small planetesimals and mergers between large planetary embryos. As a result, in order to understand the origin and evolution of the terrestrial planets’ atmospheres one needs to examine the contribution to volatile delivery and loss by both the giant impacts and planetesimal accretion. Geochemical observations of noble gases and major volatiles determine what the present day inventory of volatiles tells us about their sources and their accretion process as well as the key volatile loss mecha-nisms during Earth’s formation. In this chapter we combine geochemical observations and models of small planetesimal impacts and giant collisions to shed light on the volatile de-livery and atmospheric loss history of the Earth. We show that the data and models point to a rich interplay between volatile accretion and atmospheric mass loss facilitated by small and large impacts. This chapter is structured as follows: Earth’s geochemical constraints on the delivery of its volatiles and losses are discussed in Sect.2. Analytical impact models ranging from small planetesimal impacts to giant impacts are presented in Sect.3and used to calculate the atmospheric mass loss during accretion and the delivery of the late veneer highlighting the importance and efficiency of various different impact regimes. Conclusions follow in Sect.4.
2 Volatile Delivery to and Losses from the Earth: Constraints from
Observations
The relative abundances of a suite of elements (H, C, N, halogens, noble gases) along with their isotopic compositions allow us to decipher where volatiles were derived from during Earth’s accretion. Measurements in mantle-derived rock, combined with volatile abundances in the Earth’s exosphere, are now providing us with insights into both the total abundance of volatiles and their initial distribution on the Earth. Recent measurements are also providing key new information on the importance of processes like hydrodynamic escape and impacts in shaping the final abundance of the volatile budget for Earth (Marty2012; Halliday2013; Tucker and Mukhopadhyay2014; Sharp and Draper2013; Hirschmann2016).
2.1 Volatile Sources and the History of Volatile Delivery
Over the past fifteen years, the non-radiogenic neon (Ne) isotopes (20Ne/22Ne) have played
an important role in providing constraints on the acquisition of volatiles. Earth could have acquired Ne from potentially three sources: i) nebular Ne with 20Ne/22Ne ratio of
Fig. 1 The Neon isotopic composition of the mantle and atmosphere compared with that of the solar nebula,
solar wind irradiated material and CI chondrites. The nebular value is from Heber et al. (2012), the range of solar wind irradiated material is from Moreira and Charnoz (2016) and the CI chondrite data is from Mazor et al. (1970). The Galapagos, Iceland and Kola plume points represent the highest measured values from these plumes and are from Péron et al. (2016), Mukhopadhyay (2012) and Yokochi and Marty (2004), respectively. The MORB source is from Holland and Ballentine (2006). So far, MORBs removed from the influence of plumes have not yielded20Ne/22Ne ratios that are statistically different from 12.5. The large error bars for the CI chondrites reflect the inherent heterogeneity in Ne–Ar composition within this meteorite class. Error bars in both panels are 1 sigma and the large error bars for the CI chondrities reflect the inherent heterogeneity in Ne composition within this meteorite class
early Solar System and then incorporated into planetesimals (20Ne/22Ne of∼12.5 to 12.7,
although higher values might be possible if irradiation is not in steady state with sputter-ing; Moreira and Charnoz (2016)) and iii) Ne from carbonaceous chondrites (20Ne/22Ne of ∼8). The shallow mantle that is most frequently tapped during the mid ocean ridge
volcan-ism (MORB source) appears to have a 20Ne/22Ne of 12.49± 0.04, similar to solar wind
irradiated material that is found in gas-rich meteorites (Fig.1; Trieloff et al. (2000), Bal-lentine et al. (2005), Holland and Ballentine (2006)). However, the deep mantle, sampled by plumes, appear to reach20Ne/22Ne values close to 13.0 (Sarda et al.2000; Yokochi and Marty2004; Mukhopadhyay2012; Péron et al.2016). This value is higher than expected from solar wind irradiation of grains and significantly higher than values in carbonaceous chondrites raising the intriguing possibility that within the deepest mantle, there is a memory of Earth’s acquisition of nebular volatiles.
The presence of nebular volatiles in the deep mantle plume source suggests that volatile accretion started in the presence of nebular gases while the Earth was an embryo. While nebular gases can persists for∼10 Myrs, the median lifetime of a nebula, i.e. the time at which half of all systems lost their gas disks, is∼2.5 Myrs (Hillenbrand2008). The mean time of accretion for Earth (i.e. the time to have grown to 50% of the final mass) is∼11 Myrs (Yin et al. 2002) while the mean accretion timescale of Mars is 1.9+1.70.8 Myrs (Dauphas and Pourmand 2011; Tang and Dauphas2014). Clearly, the entire planet could not have equilibrated with the solar nebula. However, given a median lifetime of 2.5 Myrs for the nebula, a relatively high probability event would be for one or more embryos incorporated into Earth, 1–3 times the size of Mars, to have acquired nebular volatiles.
With the exception of the deep mantle plume source, signatures of nebular gases are not found elsewhere on our planet. Ne isotopes in the MORB source are similar to solar wind ir-radiated meteoritic sources (e.g. Ballentine et al.2005; Moreira2013) and the isotopic ratios of H, N, argon (Ar), krypton (Kr) and xenon (Xe) in the convecting MORB mantle source and in the atmosphere (with the exception of Xe) are most similar to chondritic meteorites
(Alexander et al.2012; Marty2012; Halliday2013; Hirschmann2016). Furthermore, even the deep mantle values are not identical to nebular values. These observations suggest that most of the nebular volatiles acquired by the Earth embryos must have been lost and volatile elements were later replenished through delivery of chondritic volatiles after the dissipation of the nebula. We note that the isotopic compositions of the major volatiles rule out comets as the dominant source of Earth’s major volatile species since these icy bodies are enriched in the heavier isotopes of H and N compared to Earth (Bockelée-Morvan et al.1998; Meier et al.1998; Jehin et al.2009; Halliday2013; Marty2012; Marty et al.2016)
The loss of nebular volatiles from within the Earth during its embryo stage may be a nec-essary consequence of the continued accretion of the Earth. After the nebula dissipates, the Earth embryo grows through impact of planetesimals and through giant impacts associated with collisions between embryos, with the Moon forming giant impact effectively terminat-ing Earth’s accretion. Since magma oceans are a consequence of giant impacts, the colli-sions between embryos would likely have released a large fraction of gases trapped within the solid Earth into the atmosphere. Hydrodynamic escape following the dissipation of the nebular and/or subsequent impacts must have led to loss of these nebular volatiles from the Earth embryo. Alternatively, some of the volatile elements may have been sequestered in the core (e.g. Dasgupta et al.2013; Hirschmann2016).
The timing of acquisition of volatiles is currently still a subject of debate (Marty2012; Halliday2013; Tucker and Mukhopadhyay2014; Hirschmann2016). The two possible hy-pothesis are that the majority of these volatiles were delivered prior to the Moon forming impact, or were delivered associated with the late accretion, defined as the last 0.3–1% of the Earth’s mass that was accreted after the formation of the Moon (e.g. Warren et al.1999; Walker2009).
Radioactive isotopes, like129I and244Pu, can provide constraints on the history of volatile
accretion. I is a volatile element while Pu is a refractory lithophile element. While the de-livery rate of Pu should be time invariant during terrestrial accretion, I dede-livery would have changed with time. Thus, the I/Pu ratio of the Earth can potentially be used as a tracer of heterogeneous volatile delivery. If mantle domains have kept a memory of the different periods of terrestrial accretion, then the heterogeneous delivery of I with respect to other refractory lithophile elements could be interrogated through the 129Xe/136∗XePu ratio of
mantle-derived basalts. Here,129Xe is produced from129I decay only during the first 100
Myrs of solar system history and136∗Xe
P uis136∗Xe produced from244Pu fission during the
first 500 Myrs of solar system history. The129Xe/136∗Xe
P uratio has been used to constrain
the closure time for volatile loss of a mantle reservoir (Pepin and Porcelli2006; Avice and Marty2014). Since129I has a shorter half-life than244Pu, for reservoirs with the same
ini-tial I/Pu ratio, higher129Xe/136∗Xe
P uratios are indicative of earlier closure to volatile loss.
The129∗Xe/136∗Xe
Puof the plume source mantle varies between 2.1± 1.6 and 2.9+0.4−0.1and
the corresponding values for MORBs are between 7.8+3.1−1.8 and 9.4+5.5−2.7(Fig.2, Mukhopad-hyay (2012), Pet˝o et al. (2013), Parai and Mukhopadhyay (2015), Caracausi et al. (2016)). If the mantle had a homogeneous I/Pu ratio, the higher129∗Xe/136∗Xe
Puratio in the MORB
source would paradoxically imply that the shallow mantle became closed to volatile loss earlier than the deep mantle. A simpler explanation is that the lower129∗Xe/136∗XePu
re-flects a lower initial I/Pu ratio for the plume source compared to the MORB source. These differences would indicate that the initial phase of Earth’s accretion was volatile poor com-pared to the later stages of accretion. This conclusion is consistent with Pd–Ag systematics (Schönbächler et al.2010) with the added distinction that the differences in the volatile ac-cretion history are still preserved in the Xe isotopic composition of the mantle. As a result, the magma oceans formed during the giant impact stage of accretion did not homogenize chemical signatures acquired during the embryo stage.
Fig. 2 Ratios of I- to Pu-derived
Xe in plume source and MORB source assuming a chondritic starting composition for Xe isotopes in the mantle (Caracausi et al.2016). The plume average was compiled from data presented in Mukhopadhyay (2012), Pet˝o et al. (2013), and Caracausi et al. (2016). The MORB average is from Parai and Mukhopadhyay (2015)
Fig. 3 Relative terrestrial
abundances of C, N, H, and halogens normalized to the chondritic relative abundances. H, C and N data are from Marty (2012) and Halliday (2013). Cl data is from Sharp and Draper (2013) and F data are from McDonough (2003) and Palme and O’Neill (2014)
The relative abundances of the volatile elements also provide key insights into when volatiles must have been delivered. For example, the C/H ratio and N/H ratio on Earth is low compared to chondrites (Fig.3), suggesting some process(es) fractionated this ratio fol-lowing delivery of chondritic volatiles. For example, while the N and H isotopic ratios on Earth are chondritic, the N/H ratio does not match any known solar system object. Therefore, delivering all of the H and N associated with late accretion presents a significant challenge. Therefore Halliday (2013) concluded that water must have been delivered during the main phase of accretion. Halliday (2013) pointed out that if the entire N budget was delivered during the late accretion, the low N/H ratio of the Earth has to imply that about 70% of the Earth’s hydrogen, in the form of water, must have been delivered prior to the Moon forming impact. Tucker and Mukhopadhyay (2014) argued that the low C/H and N/H ratio reflects modification of the chondritic relative abundances of these volatile elements on the Earth. Likewise, even though the Cl isotopic composition of Earth is chondritic (Sharp and Draper2013), the extremely low Cl/F ratio of the Earth is an indication that the Cl abun-dance must have been modified relative to the F abunabun-dance (Tucker and Mukhopadhyay
2014). Hirschmann (2016) argued that while most of the water was delivered prior to the Moon forming giant impact, since the C/S ratio of the Earth is chondritic, and because both these elements are expected to have strongly partitioned into the Earth’s core during core formation, significant C and S replenishment must have happened after cessation of core formation. That is, C and S replenishment of the Earth’s mantle and exosphere must have occurred after the formation of the Moon in association with late accretion.
Hirschmann’s (2016) argument about volatile delivery after the formation of the Moon is also compatible with the noble gases. The mantle and atmosphere have distinct non-radiogenic Ne, Kr and importantly, while the atmosphere is enriched in the heavier22Ne
iso-tope relative to the mantle, it is depleted in the heavier84Kr isotope compared to the mantle
(Holland et al.2009; Marty2012; Halliday2013). Hence, the mantle and atmosphere cannot be related through outgassing or outgassing combined with hydrodynamic escape. The sim-plest explanation for the difference in mantle and atmospheric composition would be for the
majority of atmospheric noble gases to be delivered after the last major mantle-atmosphere exchange—the Moon forming giant impact. Delivery of the noble gases associated with the late accretion, however presents a challenge. The atmospheric Kr inventory requires that the late accretion was CI chondrites, constituting∼4.5% of Earth’s mass (Marty2012), a num-ber that would lead to significant overabundances in the terrestrial budget of the platinum group elements. To resolve this, delivery of a small amount of cometary ices have been in-voked (Halliday2013; Marty et al.2016). Laboratory experiments (Bar-Nun et al.2007; Notesco et al.2003) and recent measurements of comet 67P/Churyumov-Gerasimenko (Al-twegg et al.2015; Marty et al.2016) suggest that cometary ices can trap significant amounts of noble gases. If the laboratory experiments and cometary measurements are representa-tive of ices, the budget of atmospheric noble gases could be delivered by icy planetesimals representing∼ 1% by mass of the late accretion. While the H and N isotopic compositions of comets can be significantly heavier than chondritic sources (e.g. Bockelée-Morvan et al.
1998; Meier et al.1998; Jehin et al.2009; Manfroid et al.2009; Meech et al.2009), accre-tion of such a small amount of cometary ice would lead to no detectable change in the H or N budget of the Earth. Therefore, the isotopic composition of these elements would not be perturbed away from chondritic values.
The compositions of the volatile elements therefore, suggest a complicated story of in-heritance, loss and multiple stages of replenishment. Importantly, the budget of the different volatile elements appear to have grown at different rates, governed by the balance between delivery and loss, with loss to the core or outer space controlled by geochemical behavior of the elements. Acquisition of volatile elements started in the embryo stage, evidence of which survives in the Ne, and possibly H, isotopic compositions in the deep mantle plume source. With the dissipation of the nebula, planetary growth enters the giant impact phase of accretion—a stage marked by both addition of major volatiles with chondritic composition and volatile loss from the growing Earth. During this stage, water, and most likely all of the volatile elements, were partially retained. Following, the Moon forming giant impact, C, N, S, and halogens were partially replenished through accretion of chondritic planetesimals during the late accretion phase; the atmospheric noble gases where likely acquired from the accretion of 1× 1022g of icy planetesimals, which is approximately 0.1% of the late
accre-tion mass (Marty et al.2016). This amount of icy material is not large enough to leave its mark on the abundance of the major volatiles (H, C, S, N, halogens). The delivery of the icy planetesimals could have occurred during main phase of the late accretion or at the very tail end of late accretion (e.g. Marty et al.2016).
2.2 What Were the Volatile Loss Processes and When Were Volatiles Lost? 2.2.1 Loss via Hydrodynamic Escape Versus Loss via Impacts
As discussed in Sect. 2.1, there is evidence in the deep mantle that nebular gases were acquired but signatures of nebular gases are not seen in the shallower mantle or in the atmo-sphere. This observation suggests loss of volatiles from the Earth during the main accretion phase. A simple calculation can be done based on the radiogenic Xe budget to show sig-nificant volatile loss during accretion. The abundance of 127I in the bulk silicate Earth is
5–10 ppb (Lodders and Fegley1998; Avice and Marty2014). With an initial129I/127I ratio
of 1.0× 10−4(Ozima and Podosek2002), the decay of129I to129Xe should have produced ∼1.5–3 × 1013moles of radiogenic129Xe on the Earth. The atmospheric abundance of
ra-diogenic129Xe is∼2.8 × 1011moles (Pepin1991). The mantle budget is uncertain but is
radiogenic129Xe has been lost from the Earth. The recent discovery that xenon isotopes in
the Archean atmosphere were less fractionated than modern air suggests xenon loss during the Hadean and Archean associated with a mass fractionating process (e.g. Pujol et al.2011). However, even after accounting for this mass fractionating loss,∼75% of the129Xe must
have been lost prior to ∼60 Ma, the approximate age of the Moon-forming giant impact (Avice and Marty2014). Hence, substantial xenon loss must have occurred during the giant impact phase of accretion. The significant loss of radiogenic129Xe during the giant impact
phase of accretion should not be taken to imply that all of the major volatiles were delivered after the Moon-forming giant impact. Rather, the volatile abundances could be residual to loss followed by replenishment during the late accretion phase.
Hydrodynamic escape has been invoked to explain the loss of volatiles, including Xe (Pepin1991; Pepin and Porcelli2006; Dauphas2003). Hydrodynamic loss, however, re-quires a H-rich atmosphere, such as would be present if a nebular atmosphere was captured by the proto-Earth. In fact, models for hydrodynamic escape start off with a solar composi-tion gas and explain the present abundances through loss of the H2-rich atmosphere followed
by resupply from the mantle and delivery of planetesimals during the late accretion (Pepin
1991; Dauphas2003).
The present day geochemistry of volatiles, however, shows no evidence of hydrodynamic escape. Hydrodynamic escape fractionates the isotopes as the escaping H2 preferentially
drags the lighter isotope, leaving the atmosphere enriched in the heavier isotopes. Therefore, the present day chondritic isotopic composition of the H, C, N and Cl would require the H2
flux to space to fall off at just the right time so that volatile escape is shut off exactly when chondritic isotopic composition are reached. Importantly, because the mass differences be-tween the isotopes are different for the different elements, escape for the different volatile elements has to stop at different times and precisely at the point when the isotopic composi-tions have reached chondritic ratios. Such a scenario requires an extreme fine tuning of the H2 escape flux and appears highly implausible. Alternatively, to generate the present day
isotopic compositions requires adding isotopically light volatiles sequestered in the mantle to the isotopically heavy residual atmosphere that was mass fractionated through hydrody-namic escape (Pepin1991; Pepin and Porcelli2002). However, recent measurements of Kr isotopes indicate that the mantle is actually enriched in the heavier isotopes compared to the atmosphere (Holland et al.2009). Therefore, hydrodynamic escape is not the primary loss signature recorded in the present day volatile budget. The lack of evidence for hydrody-namic escape in the present day volatile signatures does not imply that hydrodyhydrody-namic escape never occurred. Hydrodynamic escape might have taken place on embryos shortly after the nebular gas dissipated (e.g. Erkaev et al.2014). For our present day planet, it appears that if hydrodynamic escape occurred early in the accretion history (embryo stage), the resulting chemical signature has been completely overprinted by volatile resupply and additional loss mechanisms that did not fractionate the isotopes.
The process capable of driving loss without isotopic fractionation is impacts, both giant impacts as well as impacts of planetesimals. Atmospheric erosion associated with plan-etesimals has recently been shown to be quite efficient (Schlichting et al. 2015). While impact-driven loss does not fractionate the isotopes since it involves bulk ejection of the at-mosphere, impact-driven losses can fractionate the elements. This is because during impacts, water oceans and crust would be retained preferentially compared to the atmosphere. Thus, elements residing in the atmosphere (nitrogen, noble gases) would be lost in preference to those in the ocean and crust (H and C). In the rest of Sect.2.2, we will discuss constraints on magma ocean outgassing and atmospheric loss via impacts from He/Ne ratios in the mantle.
Fig. 4 3He/22Ne ratios in modern terrestrial reservoirs along with the3He/22Ne ratios in possible sources that may have contributed primordial He and Ne to the Earth. Figure after Tucker and Mukhopadhyay (2014), with solar nebula value update from Heber et al. (2012)
2.2.2 The He/Ne Ratio Difference Between MORB and Plume Sources
The3He/22Ne ratio of the present day mantle spans a large range from≤2, and possibly
down to 1, to≥10 (Fig.4, Raquin and Moreira (2009), Coltice et al. (2011), Tucker and Mukhopadhyay (2014), Péron et al. (2016)). Plumes with the most primitive 21Ne/22Ne
ratios from Galapagos and Iceland have 3He/22Ne ratios of 2 to 3 (Kurz et al. 2009; Mukhopadhyay2012; Raquin and Moreira2009) while the depleted MORB mantle has a
3He/22Ne ratio≥10 with intermediate3He/22Ne values due to mixing between the depleted
mantle and the plume source (Tucker and Mukhopadhyay2014). The3He/22Ne ratio of the
primitive reservoir sampled by plumes is higher than the nebular value by approximately a factor of∼1 to 2, with the depleted MORB source a factor of 6–10 higher than the sources of He and Ne to the Earth’s mantle (Fig.4).
Differences in the 3He/22Ne ratio between basalts from mid ocean ridges and basalts
from mantle plumes have long been recognized (e.g. Honda and McDougall1998; Moreira et al.1998; Sarda et al.2000). However, these differences were not universally accepted as reflecting a difference in the mantle sources of plumes and MORBs. Rather, magma generation and melt migration were sometimes invoked to explain the observed differences in3He/22Ne ratios between MORBs and plumes (e.g. Sarda et al.2000) with the mantle
sources of basalts assumed to have similar 3He/22Ne ratios (e.g. Trieloff and Kunz2005;
Hopp and Trieloff2008). More recent work however has unequivocally established that the mantle sources of plumes and MORBs have distinct 3He/22Ne ratios (Raquin and
Mor-eira2009; Kurz et al.2009; Füri et al.2010; Coltice et al.2011; Marty2012; Tucker and Mukhopadhyay2014). Here we focus on the new interpretations of the3He/22Ne ratio of
mantle-derived basalts over the past few years.
2.2.3 Magma Oceans and3He/22Ne Ratios
A striking feature of the3He/22Ne ratios is that the mantle reservoirs have a higher value
than even the Sun (Fig.4). How did the Earth’s interior acquire a3He/22Ne ratio higher than
even the solar nebula? To posit that the Earth’s interior is enriched in3He compared to the
Sun is unreasonable. More likely, the mantle must have preferentially lost22Ne. Based on
the factor of two higher solubility of He compared to Ne in a basaltic liquid, Honda and McDougall (1998) argued that magma ocean outgassing must have increased the MORB
source3He/22Ne ratio, as magma ocean degassing would preferentially release Ne over He
into the atmosphere. They also noted that the lower3He/22Ne of plumes might indicate a
less degassed source that was largely isolated from the MORB source for most of Earth’s history.
More recently, Coltice et al. (2011) suggested that the3He/22Ne ratio in plumes reflect
the signature of a basal magma ocean. In the aftermath of the last giant impact, the Moon-forming impact, a global magma ocean may have existed. For a peridotite liquid, the liquidus intersects the isentrope in the mid-lower mantle (e.g. Mosenfelder et al.2009; Thomas et al.
2012). Consequently, the magma ocean will first start crystallizing at the mid-lower mantle, which would ultimately isolate a layer of molten mantle above the core-mantle boundary— the basal magma ocean (BMO) (Labrosse et al.2007). As crystallization in the BMO pro-ceeds, incompatible elements, like noble gases, get enriched in the liquid. If both He–Ne partition coefficients are on order 0.01, the late crystallizing minerals could have relative high concentrations and low3He/22Ne ratios. Since iron also becomes enriched in the
resid-ual liquid as the BMO crystallizes, the late crystallizing minerals would be Fe-enriched and likely to be more dense than the surrounding mantle assemblage. As a result of its high-density, the low3He/22Ne reservoir would be entrained to low degrees in the convective
flow, preserving the reservoir over the age of the Earth. Coltice et al. (2011) noted that the factor of 5–10 higher value in the MORB is unexplained and could reflect mantle degassing, although they did not indicate when or how this degassing occurs.
Tucker and Mukhopadhyay (2014) argued that mantle outgassing associated with the long-term plate tectonic cycling is not likely to increase the3He/22Ne ratio of the mantle.
Specifically, the mechanisms they investigated for raising3He/22Ne ratios were (i) partial
melting that generates oceanic crust and a depleted mantle residue and (ii) recycling and mixing of oceanic crust and lithosphere back into the mantle. Given current determinations of partition coefficients for He and Ne (Brooker et al. 2003; Heber et al.2007; Jackson et al.2013), they found that partial melting is not an efficient process for fractionating He from Ne in the mantle. Likewise, due to ubiquitous incorporation of atmospheric noble gases, both upper and lower oceanic crust have a3He/22Ne of∼0 (Staudacher and Allègre
1988; Moreira et al.2003). If the recycling efficiency of He and Ne back into the mantle is high, recycling of oceanic crust should decrease the3He/22Ne ratio and not increase it.
Therefore, Tucker and Mukhopadhyay (2014) concluded 3He/22Ne values of 10 for the
depleted mantle could not be due to processes associated with the long-term plate tectonic cycle. They suggested that outgassing or ingassing of a magma ocean is a process that is expected to raise the mantle3He/22Ne ratio. As He is more soluble in magmas than Ne,
magma ocean ingassing or outgassing will raise the mantle3He/22Ne ratios. The differences
between the3He/22Ne ratios of plume and depleted mantle sources then reflects a difference
in deep versus shallow mantle magma ocean outgassing history of the Earth.
2.2.4 Multiple Magma Oceans, Giant Impacts and Atmospheric Losses
To explain the low3He/22Ne ratios and solar-like20Ne/22Ne ratios of ≥12.9 in plumes,
Tucker and Mukhopadhyay (2014) proposed ingassing of nebular gas from a gravitationally accreted nebular atmosphere into a magma ocean, a hypothesis that was previously proposed as the mechanism responsible for acquisition of terrestrial He and Ne (Harper and Jacobsen
1996; Mizuno et al.1980; Porcelli et al.2001; Yokochi and Marty2004). During nebular ingassing into a magma ocean, the3He/22Ne ratio of the magma ocean would be fractionated
from the nebular value of∼1.5 by the He/Ne solubility ratio of ∼ 2 to values of ∼2.3–3. The magma ocean ingassing occurs on an embryo, probably about the size of Mars or slightly larger.
Fig. 5 A proposed chronology of events to explain the highly fractionated3He/22Ne ratio of the depleted mantle. Figure after Tucker and Mukhopadhyay (2014)
The magma ocean on embryos will crystallize on a timescale on order 106to 107years
(Zahnle et al.2007; Elkins-Tanton2008; Lebrun et al.2013). The median lifetime of nebular gas is∼2.5 Myrs and the maximum observed timescale is ∼10 Myr (e.g. Hillenbrand2008). Consequently, the nebular gas will dissipate on a timescale similar to the crystallization timescale of the magma ocean. The gas in the disk has a dampening effect on planetary orbit. Consequently, dissipation of the nebular gas pumps up the eccentricity of the embryos, causing them to run into each other—the giant impact phase of terrestrial planet accretion. During the giant impact phase, magma oceans are expected as the proto-Earth grows to its present mass through violent collisions with other embryos.
Tucker and Mukhopadhyay (2014) proposed that after loss of the nebular atmosphere, and during the giant impact phase of terrestrial growth, at least two separate partial man-tle magma ocean episodes raised the3He/22Ne ratio of the shallower mantle from∼2 to ≥10, with the last magma ocean associated with the Moon-forming giant impact (Fig.5). However, during this phase, the low3He/22Ne ratio in the deep mantle must be preserved.
Thus, Tucker and Mukhopadhyay (2014) argued that the later giant impacts including the Moon forming giant impact may not have produced a whole mantle magma ocean because the timescale for turnover of a turbulently convecting magma ocean may be a short as a few weeks (Solomatov 2000; Elkins-Tanton2008; Pahlevan and Stevenson2007), and hence, the magma ocean would be expected to be chemically homogeneous. Alternatively, if whole mantle magma oceans were produced, some process must have prevented mixing within the magma ocean. One possibility to prevent such mixing might be the presence of a dense basal magma ocean prior to the impact (Nakajima and Stevenson2015).
The requirement of 2 giant impacts and magma ocean outgassing episodes comes from the He/Ne solubility ratio of ∼2 in a magma ocean (Tucker and Mukhopadhyay2014). Thus, each stage of magma ocean outgassing can only raise the residual liquid by at most a
factor of 2. The degree of outgassing of a magma ocean is, however, tied to the atmospheric boundary condition above the magma ocean. If an atmosphere that had equilibrated with a previous magma ocean remained intact during a giant impact, the new magma ocean would not outgas and fractionate3He from22Ne because it would already be in equilibrium with
the overlying atmosphere. Therefore, loss of the pre-existing atmosphere is a requirement for driving fractionation of He–Ne during magma ocean outgassing (Fig.5). The loss of the nebular atmosphere in the aftermath of magma ocean ingassing on the planetary embryo could have been achieved through hydrodynamic escape (e.g. Pepin1991), giant impacts (Genda and Abe2003,2005; Schlichting et al.2015), impacts of planetesimals (Schlicht-ing et al.2015), or a combination of the above factors. During the giant impact phase of accretion, outgassed atmospheres are expected, which are likely to be H2O–CO2 rich and
not H2-rich (e.g. Heber et al.2012). Thus, loss of an outgassed atmosphere is not likely to
be driven by hydrodynamic escape. Rather, escape is more easily ascribed to giant impacts (Genda and Abe2003) and/or impacts of planetesimals (Schlichting et al.2015).
Complete loss of the atmosphere would lead to the largest extent of fractionation, the He/Ne solubility ratio of∼2, with partial loss suppressing the extent of fractionation. Tucker and Mukhopadhyay (2014) noted that any process that leads to preferential He loss from the magma ocean or atmosphere, such as kinetically controlled degassing or hydrodynamic escape of the atmosphere, would also suppress the extent of He–Ne fractionation during de-gassing of the magma ocean. In addition, He/Ne fractionation associated with dede-gassing would also be suppressed if during the giant impact phase, meteoritic He and Ne with
3He/22Ne of∼0.9 were delivered to the Earth’s mantle. In such cases, more than the two
episodes of giant impacts may be required to drive the mantle3He/22Ne ratio to values of ∼10. The proposed chronology of events in Fig.5should therefore, be viewed as a minimum sequence of events during the formation of the Earth.
In summary, the observation of different3He/22Ne ratios in the plume and MORB mantle
reservoirs requires the formation and preservation of two distinct mantle domains during accretion (Coltice et al.2011; Tucker and Mukhopadhyay2014). High 3He/22Ne ratio of ≥10 for the MORB source suggests multiple magma oceans and multiple atmospheric loss
episodes. The lower 3He/22Ne of the plume source also appears to be associated with a
magma ocean; Coltice et al. (2011) suggested that the plume values are the signature of crystallization of a basal magma ocean while Tucker and Mukhopadhyay (2014) associated it within ingassing of nebular gases into a magma ocean on the proto Earth. Tucker and Mukhopadhyay (2014) argued that given the He/Ne solubility ratio of 2, the low3He/22Ne
ratios in plumes is not likely to have its origin in a basal magma ocean produced by the last giant impact (the Moon-forming giant impact) because that magma ocean must have produced the high3He/22Ne ratio now sampled in the depleted mantle. However, a basal
magma ocean sequestering low3He/22Ne ratios in the deep mantle could have formed from
a previous whole-mantle magma ocean episode and recent work suggests that a dense basal magma ocean layer may not get mixed in with the rest of the mantle in the aftermath of a giant impact (e.g. Nakajima and Stevenson2015).
2.2.5 Terrestrial Volatile Signatures and Atmospheric Loss
Geochemical evidence for multiple magma oceans associated with multiple giant impacts is consistent with dynamic simulations of terrestrial planet formation that suggest more than just the Moon-forming giant impact (e.g. Chambers and Wetherill1998; Chambers2001). The atmospheric loss episodes put forth to explain the fractionation of He/Ne during magma ocean outgassing, also provide explanations for other features of the Earth’s volatile signa-tures, such as the depletion of carbon, and the very large depletion of nitrogen, with respect
to water (Fig.3). Carbon depletion has previously been linked to atmospheric loss or to parti-tioning of carbon to the core (Hirschmann and Dasgupta2009; Hirschmann2016). Likewise, N depletion has been linked to core formation (Marty2012). However, since carbon appears to be more siderophile than nitrogen by factors of 25 to 100, depletion of N by core forma-tion is unlikely because it would be accompanied by a significantly greater depleforma-tion of C (Tucker and Mukhopadhyay2014; Hirschmann2016). The observed moderate depletion in C and extreme depletion in N might be explained due to atmospheric loss associated with impacts. Carbon can form bicarbonate and carbonate ions and as a result, a significantly larger fraction of C can be the ocean (or crust) compared to N. Since both giant impacts and impacts of planetesimals preferentially remove the atmosphere over an ocean (Genda and Abe2005; Schlichting et al. 2015), atmospheric loss would lead to large fractionation in the N/H ratio and a more modest fractionation in the C/H ratio (Fig.3). Since impacts lead to bulk ejection of the atmosphere, the C and N isotopic composition of the Earth would remain chondritic.
Atmospheric loss may also explain why chlorine is depleted in the Earth relative to flu-orine (Fig. 3). Compared to Cl, F is more soluble in magmas by a factor of 4 (Andrews et al.2009). Consequently, magma ocean degassing would lead to preferential transport of Cl to the Earth’s surface compared to F. If chlorine is lost due to giant impacts or impacts of planetesimals, it would explain the depletion of Cl with respect to F. On the Earth’s sur-face, chlorine would be expected to reside primarily in the ocean and crust unless a steam atmosphere was present. Thus, loss of chlorine might also require loss of water, either in the form of partial ocean loss or through erosion of a steam atmosphere. Overall, Earth’s volatile budget records the violent events during accretion. While late accretion contributed volatiles to Earth (e.g. Albarède2009; Marty2012; Hirschmann2016), late accretion did not overprint the volatile characteristics acquired during the main stages of Earth’s accretion (Halliday2013; Tucker and Mukhopadhyay2014; Dauphas and Morbidelli2014).
3 Atmospheric Loss by Small and Large Impacts
Terrestrial planet formation consists of successive mergers of small planetesimals and larger protoplanets accumulating into final planets ranging in size from Mars to Earth. The de-livery of volatiles and atmospheric erosion are intricately linked to planetary growth since planetesimal impacts and giant impacts can both deliver volatiles and lead to their loss due to atmospheric erosion.
We present here simple analytic models which we use to calculated the atmospheric mass loss due to impacts during planet formation. The aim of these models is to distill the somewhat complex and complicated nature of impacts to their essential physics, which allows us to gain an intuitive understanding of atmospheric mass loss results and to apply them over the whole range of possible impactor sizes. These models are not suitable to determine the precise outcome of a single specific impact, which for example have been investigated by Shuvalov (2009) and Liu et al. (2015), but capture the overall mass loss results in an averaged sense, e.g. once averaged over various impact geometries and angles. For the purpose of this chapter we assume that the atmosphere is, to first order, isothermal such that its density profile is exponential and given by
ρ= ρ0exp[−z/h], (2)
where ρ0 is the density on the ground, z the height in the atmosphere measured from the
Fig. 6 Illustration of the impact
geometry. Planetesimal impacts can only eject atmosphere locally. Treating their impact as a point-like explosion leading to an isotropic shock at the impact site, the maximum atmospheric mass that they can eject in a single impact is given by all the mass above the tangent plane, which is a fraction given by h/2R of the total atmospheric mass. Figure after Schlichting et al. (2015)
Fig. 7 Illustration of a giant
impact. 1) The giant impact ejects atmosphere and ejecta close to the impact point and launches a strong shock. 2) The shock front propagates through the target causing a global ground motion. 3) This ground motion in turn launches a strong shock into the planetary atmosphere, which can lead to loss of a significant fraction of or even the entire atmosphere. Figure after Genda and Abe (2003) and Schlichting et al. (2015)
the results derived here will change for an adiabatic atmosphere in which the heat transport is facilitated by convection. We assume here that the atmosphere is planar, which is valid for the terrestrial planets since their atmospheric scale heights, h, are small compared to their radii, R. For the current Earth, h/R∼ 0.1% so the planar condition is indeed well justified for the terrestrial planets. In contrast, for close in exoplanets the planar condition breaks down and full spherical treatment should be used, see for example Inamdar and Schlichting (2015,2016).
Impacts can lead to atmospheric losses in two distinct ways: First, the expansion of plumes generated at the impact site can erode the atmosphere locally, but not globally (see Fig.6). Such loss can be caused by small planetesimal impacts and is discussed in detail immediately below. Second, when the target and impactor are comparable in mass, a giant impact can create a strong shock that transverses through the entire planet leading to global ground motions that in turn can launch a shock into the atmosphere (see Fig.7). The shock accelerates the atmosphere, and fluid parcels that are accelerated to above the escape veloc-ity of the target are lost. A simple model calculating the atmospheric mass loss due to giant impacts is described in Sect.3.3.
3.1 Atmospheric Loss by Small Impacts
Small impacts can individually only eject planetary atmospheres locally, close to their im-pact sites. However, collectively they play an important role in atmospheric loss during planet formation, since they are many orders of magnitude more efficient per unit impactor mass in atmospheric erosion than giant impacts (Schlichting et al.2015).
3.1.1 Mass Loss Regimes
The first calculations of atmospheric loss by planetesimal impacts used the Zel’dovich and Raizer (1967) solution for the expansion of a vapour plume and compared the momentum of the expanding gas and the mass of the overlying atmosphere (e.g. Melosh and Vickery1989; Vickery and Melosh1990; Ahrens1993). Here we use a similar approach and compare our analytic results to atmospheric mass loss results from numerical simulations. When the Earth or a protoplanet is hit by a small impactor, the impactor’s velocity is rapidly decelerated and its kinetic energy is converted into heat and pressure leading to something analogous to an explosion. We model the impact as an isotropic point-like explosion on the surface of the target. Assuming momentum conservation and impact velocities comparable to the escape velocity, vesc, yields an ejecta mass comparable to the impactor mass, mImp, that propagates
isotropically with a velocity of order vescinto a half-sphere centered on the impact site. This
implies that atmosphere is only lost where its mass per unit solid angle, mθ, as measured
from the impact site, is less than that of the ejecta, which is equal to mImp/2π .
More generally we find for an arbitrary impact velocity and momentum transfer effi-ciency in the impact, η, that atmosphere is only ejected within a solid angle, θ , where
mθ≤ η vImp vesc mImp 2π . (3)
For the density profile given in (2) we have
mθ= 2πρ0
∞
0
exp−a2+ 2aR cos θ/2Rha2da, (4) where z is the height in the atmosphere above the ground and it is related to a, the distance from the impact site to the top of the atmosphere (see Fig.6), by z= (a2+ 2aR cos θ)/2R.
Note, the integration in (4) is only over a and not θ since the explosion at the impact site is assumed to be isotropic (Schlichting et al.2015). This leads to two interesting limits: 1) There is a lower limit on the impactor size that can eject any atmosphere.
2) There is a lower limit on the impactor size that can eject all the atmospheric mass above the tangent plane.
The first limit can be derived by equating the mass residing in the vertical column above the impact site to (3). Integrating (4) for θ= 0 yields a minimum impactor mass of
mmin= 4πρ0h3η−1 vesc vImp , (5)
which reduces to the result obtained by Schlichting et al. (2015) for vimp∼ vescand η∼ 1,
The second limit is obtained by integrating (4) for θ= π/2 which yields the minimum impactor mass that can eject all the mass above the tangent plane, mcap, and is given by
mcap= √ 2ρ0(π hR)3/2η−1 vesc vImp , (6)
which again reduces to the result obtained by Schlichting et al. (2015) for vImp∼ vescand
η∼ 1, and yields rcap= (ρ0/ρimp)1/3(hR)1/2∼ 25 km for the current Earth (Fig.8c). All
impactors above this size will eject all the atmospheric mass above the tangent plane but not more than that, unless they are so large that they are in the giant impact regime which is discussed in Sect.3.3.
3.1.2 Mass Loss Efficiencies
Next we turn to calculating the mass loss efficiencies for the various impactor sizes. The mass loss efficiency is given by the amount of atmospheric mass lost divided by the impactor mass needed to achieve the loss. These two quantities are not equal because we model the explosion at the impact site as isotropic whereas the atmospheric mass follows an isothermal or adiabatic density profile (see (2)). As a result the atmospheric mass directly vertically above the impact site is easiest to eject, since it has the lowest column density and the atmospheric mass along the tangent plane is hardest to eject since it has the largest column density. The atmospheric mass residing inside a cone subtended by an angle θ measured from the normal to the impact site (see Fig.8) is
MEject,θ= 2πρ0 a=∞
a=0 θ =θ
θ =0
exp[−z/h] sin θ a2dθ da. (7) The mass loss efficiency is given by dividing (7) by (4) and evaluating it for various impactor sizes. Figure9shows the mass loss efficiency as a function of impactor size. Small impactors with r∗=√3rminare the most efficient impactors per unit mass in ejecting the atmosphere.
For the current Earth this corresponds to bodies with r∼ 2 km. The value ofMEject/mImp
decreases rapidly for larger planetesimals because the atmosphere along the tangent plane is harder to eject, due to its larger column density. In addition, once impactors are larger than
rcap, each impactor only ejects the mass residing above the tangent plane, which reduces the
mass loss efficiency per unit impactor mass.
Integrating over the whole cap, i.e. from θ= 0 to θ = π/2, yields a total cap mass of
Mcap= 2πρ0h2R, (8)
in the limit that R h, which applies for the terrestrial planets. This is the maximum atmo-spheric mass that a single planetesimal impact can eject and is given by all the mass above the tangent plane of the impact site. The ratio of the mass in the cap compared to the total atmospheric mass is thereforeMcap/Matmos= h/2R.
3.1.3 Impactor Size Distribution
Figure9shows the ratio of atmospheric mass ejected to impactor mass. To understand the atmospheric mass loss over time we calculate the atmospheric loss due to a population of planetesimal impactors of various sizes. The impactors are assumed to follow a power-law size distribution such that the cumulative impactor flux can be parametrized as N (> r)=
Fig. 8 Different planetesimal
impact regimes. The size of the shaded hemisphere is scaled to the impactor mass (assuming an impact velocity comparable to the escape velocity) such that the mass per unit solid angle in the hemisphere presented by the shaded region is given by
mImp/2π as measured radially
outward from the impact site. Panel a) illustrates the limit in which the impactor is able to only eject the atmosphere vertically above the impact site, which corresponds to mmin= 4πρ0h3. Panel b) corresponds to an intermediate regime, where the impactor is able to eject all the atmospheric mass residing in the red cone. Panel c) displays the limit in which an impactor is large enough to eject all the atmospheric mass residing above the tangent plane of the impact site, which corresponds to a minimum impactor mass of
mcap= √
2ρ0(π hR)3/2
N0(r/r0)−q+1, where q is the differential power law index and N0is normalized to impactors
of size r0and is the impactor flux (number per unit time per unit area). The atmospheric mass
loss rate is then given by
dMatmos dt = −πR 2N0(q− 1) r0 rmax rmin r r0 −q MEject(r)dr, (9)
Fig. 9 Ratio of atmospheric mass ejected to impactor mass,MEject/mImp. Numerical values are scaled
to the current Earth’s atmosphere and shown for vimpη/vesc∼ 1. Small impactors with r∗=√3rminare the
most efficient impactors per unit mass in ejecting the atmosphere and the ejection efficiency decreases rapidly for larger planetesimals. Whether or not planetesimal impacts will lead to a net loss of planetary atmospheres depends on the impactor sizes distribution, their volatile budget and the amount of outgassing their impacts can initiate. The three dotted horizontal lines correspond to volatile contents of 5 wt.% (representative of some of the most water rich carbonaceous chondrites), 0.05 wt.% (representative of the average water content in the bulk Earth excluding the hydrosphere) and 0.0005 wt.% corresponding to an estimate of the minimum water content of the bulk moon (McCubbin et al.2010). For comparison, data from oblique impact simulations for escape velocities of 11.2 km/s and impact velocities of 30 km/s from Shuvalov (2009) are shown by the orange points. Figure after Schlichting et al. (2015)
whereMEject(r)is the atmospheric mass ejected due to an impactor with radius r . Solving
(9) yields Matmos(t )= M0 1− t t∗ 3/(q−1) , (10)
where M0is the initial atmospheric mass at t= 0 and t∗is the time it takes to erode the entire
atmosphere (Melosh and Vickery1989). Equation (9) shows that the whole atmosphere is eroded in a finite time, which is due to the fact that as the atmosphere is lost, its density decreases and even smaller impactors, which are more numerous, start to contribute to the mass loss. For 1 < q < 3, the value for t∗is
tq<3∗ = 6 π(q− 1)CRhN0 π h 8R M0 m0 (q−1)/3 , (11) and for q > 3 it is tq>3∗ = 3(q− 3) π(q− 1)h2N0 h R 2 M0 m0 (q−1)/3 , (12)
where m0= 4πρimpr03/3 (Schlichting et al.2015). We get two different expressions for t∗
de-pending on the power-law index of the planetesimal size distribution because for 1 < q < 3 atmospheric mass loss is dominated by impactors that remove the whole atmospheric mass above the tangent plane and for q > 3 the atmospheric mass loss is dominated by im-pactors that each only eject part of the atmospheric mass residing above the tangent plane.
Fig. 10 Atmospheric mass loss over time for three different initial atmospheric masses normalized to the
current Earth’s atmosphere. The impactor population contains a total mass of 0.1% Earth’s masses, consistent with the mass inferred for the late veneer, and has a power-law size distribution with a differential power-law index of q= 2.8 as inferred from the lunar cratering record by Neukum et al. (2001). The largest impactor is 300 km in size and the smallest 1 m. The impactor flux is assumed to be uniform over 4.5 Gyrs, but the total atmospheric mass loss would remain the same for a varying impactor flux over time as long as the total impactor mass remains unchanged. When calculating the mass loss efficiency we assumed vimpη/vesc∼ 1.
The thick-red dashed lines are from numerical Monte Carlo simulations that calculate the atmospheric loss due to each individual impact and the thin black lines are the analytic solutions given in (10)
The constant C in (11) parameterizes any additional contribution to the atmospheric loss by impactors that are smaller than those that eject all the mass above the tangent planet. Melosh and Vickery (1989) investigated the mass loss due to impactors that remove all the atmosphere above the tangent plane (i.e. the 1 < q < 3 regime) but they assumed that Mcap= mcap (instead ofMcap= mcap(2h/π R)1/2 see (6) and (8)), ignoring the fact that
the atmospheric column density is larger in the direction along the tangent plane than di-rectly vertically above the impact site, and they also neglected the numerical coefficient C discussed above.
Figure10shows atmospheric erosion over time due to an impactor size distribution with
q= 2.8 and an impactor flux that bombards the Earth uniformly over 4.5 Gyrs with a total
of 0.1% of the mass of the Earth. The largest body is 300 km and the smallest 1 m in radius. The thin black lines correspond to the analytic solution given in (10) and the red dotted lines to numerical simulations that calculate the atmospheric loss after each impact and determine the evolution of the atmospheric mass over time. As expected, the numerical and analytical results are in excellent agreement. They demonstrate that small impactors that contain about 0.1% of the mass of the Earth, which is about the mass inferred for the late veneer, can erode the entire Earth’s atmosphere, if the Earth’s initial atmospheric mass was similar to its current one (Schlichting et al.2015). Whether such a bombardment by small impactors leads to a net loss or gain in atmospheric mass depends on the volatile content of the impactors themselves, how much of their volatiles end up in the atmosphere and the amount of volatiles released from the impact melt pools that are created by the impacts.
3.2 Atmospheric Loss by Giant Impacts
Giant impacts are believed to be the last major assembly stage of terrestrial planet formation (e.g. Chambers2001) and may also play a major role in the formation of close-in exoplan-ets that have masses between that of Earth and Neptune (e.g. Raymond et al.2008; Hansen and Murray2012; Inamdar and Schlichting2015). The atmospheric survival during a giant impact has been examined by several groups both in the context of terrestrial planet for-mation (Genda and Abe2003,2005; Stewart et al. 2014; Schlichting et al.2015) and in that of close-in exoplanets (Inamdar and Schlichting2015,2016). These works integrate the hydrodynamic equations of motion of the planetary atmosphere and determine the amount of atmospheric mass loss for a given ground velocity that launches a strong shock into the atmosphere. Recently Liu et al. (2015) presented the first results of three dimensional hy-drodynamic simulations of atmospheric mass loss that model the giant impact as well as the atmospheric mass loss together. Here we calculate the atmospheric loss due to a given ground motion. We assume that the atmosphere is planar. In addition we assume that radia-tive cooling can be neglected such that the flow is adiabatic. The hydrodynamic equations describing the flow are then given by
1 ρ Dρ Dt + ∂u ∂z = 0, (13) Du Dt + 1 ρ ∂p ∂z = 0, (14) 1 p Dp Dt − γ ρ Dρ Dt = 0, (15)
where γ is the adiabatic index and D/Dt the ordinary Stokes time derivative. The solutions to the hydrodynamic equations above can be separated into their time-dependent and spatial parts allowing for analytic self-similar solutions (Raizer 1964; Grover and Hardy1966; Schlichting et al.2015). Specifically we have,
ρ(z, t )= ρ0exp −Z(t)/hG(ζ ), u(z, t )= ˙ZU(ζ), p(z, t )= ρ0exp −Z(t)/h ˙Z2P (ζ ) (16)
where Z(t) is the position of the shock front and ζ= (z − Z(t))/h. The similarity variables for the density, velocity and pressure are given by G(ζ ), U (ζ ) and P (ζ ), respectively. Using the expressions in (16) and substituting them into the hydrodynamic equations (13)–(15) yields for the spatial parts
1 G dG dζ(U− 1) + dU dζ = 1, (17) (U− 1)dU dζ + 1 G dP dζ = − U α, (18) (U− 1) 1 P dP dζ − γ G dG dζ = −2 α − γ + 1, (19)
and a time dependent part given by
˙Z2
Fig. 11 Mass loss fraction,
χloss, as a function of vg/vescfor
an adiabatic atmosphere in black and an isothermal atmosphere shown as red-dashed line. Both loss curves correspond to and adiabatic index of γ= 5/3
Using the strong shock conditions we have
G(0)=γ+ 1 γ− 1, U (0)= 2 γ + 1, P (0)= 2 γ + 1. (21)
We refer the reader to Schlichting et al. (2015) for a full derivation of the self-similar solu-tions to Equasolu-tions (17)–(20). Figure11shows the atmospheric mass loss fraction as a func-tion of the ground velocity with which the shock is launched into the atmosphere, where the atmospheric mass loss fraction is defined as
χloss= exp[−zesc/ h] (22)
where zescis the initial height in the atmosphere of the fluid element that has been accelerated
to a velocity equal to the escape velocity by the shock, such that the atmosphere at z≥ zesc
Fig. 12 Illustration of the impact
geometry. An impactor of mass,
m, and impact velocity, vimp,
impacts a target with mass, M , and radius, R. Assuming momentum conservation, we calculate the shocked fluid velocity, vs, and the component
of the ground velocity normal to the surface, vg, as a function of
the distance from the impact point. Figure after Schlichting et al. (2015)
3.2.1 Global Mass Loss
Above we calculated the atmospheric mass loss due to a given ground velocity. In order to calculate the total global atmospheric mass loss, we need to relate the ground velocity to the impactor’s mass and impact velocity. We do this by using a simple model following Schlichting et al. (2015). We approximate the impact as a point like explosion on the planet’s surface, which results in a self-similar solution of the second type (Zel’dovich and Raizer
1967). As the shock propagates through the planet, its velocity must fall off faster than dictated by energy conservation but slower than required by momentum conservation. This is because as the shock propagates it must lose energy, but its momentum is increased by the nonzero pressure in the target. Catastrophic impact simulations find scaling laws that are close to ones resulting from momentum conservation (Love and Ahrens 1996; Benz and Asphaug1999; Leinhardt and Stewart2009). Assuming momentum conservation and a constant density target, the velocity of the shocked fluid traveling through the protoplanet is
vs= vImp m M 1 (l/2R)3(4− 3(l/2R)), (23)
where l is the distance of the shock travelled from the impact point, (see Fig.12). The ground velocity with which the shock is launched into the atmosphere is due to the component of the shocked fluid velocity that is perpendicular to the planet’s surface, such that vg= vsl/(2R),
which yields vg= vImp m M 1 (l/2R)2(4− 3(l/2R)). (24)
Using this ground velocity we can now calculate the global atmospheric mass loss by sum-ming the loss from all the location on the planet for the various ground velocities. The resulting atmospheric loss as a function of (vImp/vesc)(m/M)for an isothermal atmosphere
is shown in Fig.13. The total atmospheric loss consists of two components: The first is from the area of the planet’s surface where the ground motion is large enough such that locally all the atmosphere is lost (dashed line in Fig.13), the second component corresponds to the region of the planet where the local ground velocity is small enough such that only part of the atmosphere is lost (thin solid line in Fig.13) (Schlichting et al.2015). Independent of the exact value of the adiabatic index, we find that the global atmospheric mass loss fraction is for an isothermal atmosphere well approximated by
Xloss= 0.4 vImpm vescM + 1.4 vImpm vescM 2 − 0.8 vImpm vescM 3 , (25)
Fig. 13 Global mass loss
fraction (thick solid line), calculated by taking into account the different ground velocities across the planet’s surface. The total atmospheric loss consists of two components: the first is from the area of the planet’s surface where the ground motion is large enough such that locally all the atmosphere is lost (dashed line) and the second component corresponds to the regions of the planet’s surface where the local ground velocity is too small to lead to complete loss but large enough such that a fluid parcel residing higher in the atmosphere is accelerated to velocities exceeding the escape velocity such that the upper parts of the atmosphere are lost (thin solid line). The analytic fit given by the equation in the top panel is shown as dotted line. Figure after Schlichting et al. (2015)
whereas for an adiabatic atmosphere we find
Xloss= 0.4 vImpm vescM + 1.8 vImpm vescM 2 − 1.2 vImpm vescM 3 . (26)
In both cases, Xloss 0.4(m/M)(vImp/vesc)in the limit that (vImp/vesc)(m/M) 1. We find
a mass loss fraction of 6% for an Mars-sized impactor hitting an 0.9M⊕protoplanet with
vimp∼ vesc, which is about a factor of 2 lower than estimates by Genda and Abe (2003) who
assumed a single averaged ground velocity of 4–5 km/s for all locations on the planet. The presence of an ocean can significantly enhance the atmospheric mass loss (Genda and Abe
2005). Lock et al. (2014) find that with a large ocean up to 30% of the atmosphere can be lost in the canonical Moon-forming impact.
3.3 Comparison of Atmospheric Loss Across All Impact Regimes
We now turn to comparing the three different atmospheric mass loss regimes derived above. First we turn to the regime, where the mass loss is due to small impactors that can only
